MATH 512, FALL 14 COMBINATORIAL THEORY WEEK 1

Let κ be a regular (i.e. cf(κ) = κ) uncountable cardinal. Definition 1. A set C ⊂ κ is closed and unbounded (club) in κ if • for all α < κ, there is β ∈ C \ α, and • for all increasing sequences hαi | i < τi of ordinals in C for some τ < κ, supi<τ αi ∈ C. Some examples: κ \ α, for all α < κ; the set of limit ordinals less than κ. Lemma 2. Suppose that C,D are clubs in κ. Then so is C ∩ D.

Proof. To show closure, suppose that τ < κ and hαi | i < τi is an increasing sequence of ordinals in C ∩ D, and let α = supξ<τ αξ. Since C is closed, we have that α ∈ C. Since D is closed, we have that α ∈ D. To show unboundedness, fix α < κ. Let α0 > α be a point in C. Then let β0 > α0 be a point in D. Continue inductively, to build increasing sequences hαn | n < ωi of points in C and hβn | n < ωi of points in D, such that for all n, αn < βn < αn+1. Then β := supn αn = supn βn ∈ C ∩ D, and β > α.  Definition 3. A set S ⊂ κ is stationary if for all clubs C, S ∩ C 6= ∅ κ Some examples: every club set, Eω := {α < κ | cf(α) = ω}. Also note that if S is stationary and C is a club, then S ∩ C is stationary.

Definition 4. Let hAξ | ξ < κi be subsets of κ. The diagonal intersection is T defined to be 4ξ<κAξ := {β < κ | β ∈ ξ<β Aξ}. A filter is called normal if it is closed under diagonal intersections. Proposition 5. T (1) If hCξ | ξ < τi are clubs in κ for some τ < κ, then ξ<τ Cξ is also a club. (2) If hCξ | ξ < κi are clubs in κ, then 4ξ<τ Cξ is a club. The club filter on κ is the collection of all of κ containing a club. If follows by the above that the club filter is a normal κ-complete filter on κ, and it contains all complements of bounded sets. Theorem 6. (Fodor) Suppose that S ⊂ κ is stationary and f : S → κ is a regressive function, i.e. f(α) < α for all α ∈ S. Then there is a stationary T ⊂ S, such that f is constant on T . Proof. Otherwise, for all γ < κ, f −1(γ) is nonstationary, i.e. there is a club −1 Cγ with Cγ ∩ f (γ) = ∅. Let C = 4γ<κCγ, and let α ∈ C ∩ S. Set 1 2 MATH 512, FALL 14 COMBINATORIAL WEEK 1

γ := f(α). Since f is regressive, γ < α, and so by the definition of diagonal −1 intersection, α ∈ Cγ. But α ∈ f (γ). Contradiction.  The conclusion of this theorem is actually a necessary and sufficient con- dition for normality. An application of Fodor’s theorem is the following fact:

Fact (Solovay): Every stationary S of κ can be partitioned into κ many disjoint stationary subsets. (for the proof, see Chapter 8 of Jech) Definition 7. A cardinal κ is inaccessible if it is regular and strong limit (i.e. τ < κ → 2τ < κ). Proposition 8. Suppose κ is inaccessible. Then the set of cardinals below κ is club. Definition 9. An inaccessible cardinal κ is Mahlo if the set of regular car- dinals below κ is stationary. So far we have defined clubs and stationary subsets of a cardinal. For an ordinal α, c ⊂ α is a club in α is it is closed and unbounded in α, and s ⊂ α is stationary in α is it meets every club in α. For a set B ⊂ β, lim(B) will denote the limit points of B, i.e. lim(B) := {α < β | B ∩ α is unbounded}. Note that if B is unbounded, then lim(B) is a club. Also, for any club C, lim(C) ⊂ C. Definition 10. Let S ⊂ κ be stationary. S reflects if for some α < κ, S ∩ α is stationary in α. Definition 11. For a stationary set T , Refl(T ) denotes the statement that every stationary subset of T reflects. κ For example, for any uncountable regular κ, Eω reflects. Proposition 12. Let κ be any cardinal (possibly singular), and let T be a κ+ stationary subset of Eκ . Then T does not reflect. Proof. Let α < κ+ be any point. Let C ⊂ α be club in α with o.t.(C) = cf(α) ≤ κ. Then lim(C) is a club subset of α, disjoint from T .  + Definition 13. κ asserts the existence of a sequence hCα | α < κ i, such that for every α,

• Cα is club in α with o.t.(Cα) ≤ κ; • if β ∈ lim(Cα), then Cα ∩ β = Cβ. + Lemma 14. κ implies ¬Refl(S) for every stationary S ⊂ κ . + Proof. Suppose that hCα | α < κ i is a square sequence and S is a stationary + subset of κ . Let F (α) := o.t.(Cα). By Fodor, there is a stationary subset T ⊂ S, such that F is constant on T . I.e. for some δ, for all α ∈ T , o.t.(Cα) = δ. We claim that T does not reflect. For otherwise, if T ∩ α is MATH 512, FALL 14 COMBINATORIAL SET THEORY WEEK 1 3 stationary in α, then T ∩ lim(Cα) is also stationary in α. Let β < γ be two points in T ∩ lim(Cα). Then since Cα ∩ γ = Cγ, we have that β ∈ lim(Cγ), and so Cγ ∩ β = Cβ. It follows that F (β) = o.t.(Cβ) < o.t.(Cγ) = F (γ). Contradiction with β, γ ∈ T .  Next we give some weakenings of square: + Definition 15. κ,λ asserts the existence of a sequence hCα | α < κ i, such that for every α,

• 1 ≤ |Cα| ≤ λ, • for every α, for every C ∈ Cα, C is club in α with o.t.(C) ≤ κ; • for every α, for every C ∈ Cα, for every β ∈ lim(C), we have C ∩β ∈ Cβ. ∗ The principle weak square is κ := κ,κ. We have that for any λ < κ, ∗ κ → κ,λ → κ. <κ ∗ Lemma 16. Suppose that κ = κ. Then κ holds. + Proof. For every limit α < κ with cf(α) < κ, let Cα := {C ⊂ α | C is a club, |C| < κ}, i.e. all club subsets of α of size less than κ. Since <κ κ = κ, we have that |Cα| = κ. + If κ is regular, for every limit α < κ with cf(α) = κ, let Cα be any club in α of order type κ. Set Cα = {Cα}. Suppose that C ∈ Cα and β ∈ lim(C). Then since |C| ≤ κ, we have that cf(β) < κ and C ∩ β is a club subset of β of size less than κ. So, by definition, C ∩ β ∈ Cβ.  Note that the above implies that weak square holds for all inaccessible κ. Also, under GCH, weak square will hold for every regular κ. So, we will be ∗ most interested in κ when κ is singular. In general square principles are a “incompactness” type properties: a property that a structure lacks, but all of its substructures of smaller cardi- nality have. This is examplified in the following lemma: + Lemma 17. Suppose that hCα | α < κ i is a κ,λ sequence, for some 1 ≤ λ ≤ κ. Then there is no club C ⊂ κ+, such that for all α ∈ lim(C), C ∩ α ∈ Cα. Proof. If C is a club in κ+, let α ∈ lim(C) be such that o.t.(C ∩ α) > κ. + We can always find such an α, since o.t.(C) = κ . But for every E ∈ Cα, o.t.(E) ≤ κ, so C ∩ α∈ / Cα.  Definition 18. A tree (T, <) is a partially ordered set, such that for every x ∈ T , the set of predecessors of x, pred(x) := {y ∈ T | y < x} is well ordered by <. We set: • for x ∈ T , level(x) := o.t.(pred(x)), 4 MATH 512, FALL 14 COMBINATORIAL SET THEORY WEEK 1

• the height of the tree, ht(T ) := sup{level(x) + 1 | x ∈ T }, • for α < ht(T ), the α-th level of T is Tα := {x ∈ T | level(x) = α}. We also say that b ⊂ T is a branch, if b is a maximal linearly ordered subset of T .

Note that if b is a branch, then for every level α, |b ∩ Tα| ≤ 1, and if β < α, then by maximality b ∩ Tα 6= ∅ implies that b ∩ Tβ 6= ∅. We say that b is unbounded (or cofinal) if for all α < ht(T ), b ∩ Tα 6= ∅. Definition 19. The tree property holds at κ, for a regular cardinal κ, if every tree of height κ and levels of size less than κ, has an unbounded branch. We denote this by TPκ. Below we list some facts about the tree property: (1) (K¨onig)TP holds at ω. (2) (Aronszajn) TP fails at ω1. (3) TP can hold at ω2 (and above), assuming some large cardinals. ∗ + (4) κ implies that the tree property fails at κ . Definition 20. An inaccessible cardinal κ is weakly compact if it satisfies the tree property. There are several equivalent definitions of a weakly compact. We list two of them for completeness:

(1) κ is weakly compact iff κ is inaccessible and Lκ,ω satisfies the Weak Compactness Theorem. Here the language Lκ,ω contains conjunctions and disjunctions of size less than κ. It satisfies the weak Compactness Theorem if for every set of sentences Σ ⊂ Lκ,ω with |Σ| ≤ κ, if every S ⊂ Σ with |S| < κ has a model, then Σ has a model. (2) κ is weakly compact iff every function F :[κ]2 → 2, there is a set H ⊂ κ of size κ such that F is constant on H. Such a set is called homogeneous.

It turns out that TP at ω2 is equiconsistent with the existence of a weakly compact cardinal. More precisely: Theorem 21. (Mitchell) If κ is weakly compact, then there is a forcing extension, in which κ is ℵ2, and the tree property holds at ℵ2.

Theorem 22. (Silver) ℵ2 has the tree property, then ℵ2 is weakly compact in L. Later in the course we will go over the proof of Mitchell’s theorem. hat measurable cardinals are weakly compact.